Geometry of manifold of special holonomy and gauge theory/gravity correspondence

超弦理论中的D膜为研究规范理论的数学结构提供了一种新的方法。例如，例外完整群G_2，自旋(7)的流形中的D-膜描述了超对称规范理论的非微扰动力学。与Y.Yasui合作，我们构造了与SU(3)/U(1)主轨道同调的自旋(7)完整的显式度规。特别地，我们发现了一类新的度量，它渐近具有一个有限半径的圆S^1。这是四维引力瞬子的高维模拟；Taub-NUT Mtic和Atiyah-Hitchin Mtic。我们期望这些度规在M理论紧致化方面有一定的应用。弦理论的目标之一是规范理论和引力的统一，规范理论/引力对应最近引起了人们的广泛关注。当我们考虑拓扑规范/弦理论时，对应关系可以用更严格的数学方法来表示。奈克拉索夫最近的重要成果之一是四维规范理论中的瞬子计数。与T.Eguchi合作，我们证明了Nekrasov的瞬子计数配分函数可以作为拓扑弦理论的配分函数重现。SU(N)Seiberg-Witten理论的配分函数是通过考虑目标空间是P^1上的A型A_(N-1)纤维的拓扑弦而得到的。物质场可以通过在目标空间上爆破来并入。计算中使用的方法与表示论、组合学、纽结理论和可积系统等数学分支有着密切的关系。

D-branes in superstring theory provide a new method of investigating the mathematical structure of gauge theory. For example, D-branes in manifolds of exceptional holonomy group G_2,Spin(7) describe non-perturbative dynamics of supersymmetric gauge theory. In collaboration with Y.Yasui, we constructed explicit metrics of Spin(7) holonomy, that are of cohomogeneity one with SU(3)/U(1) principal orbit. In particular we found a new type of metrics that has asymptotically a circle S^1 of finite radius. This is a higher dimensional analog of four dimensional gravitational instantons ; Taub-NUT metic and Atiyah-Hitchin metic. We expect these metrics have some applications to M theory compactification.One of aims of string theory is a unification of gauge theory and gravity and gauge theory/gravity correspondence has been attracted much attention recently. When we consider topological gauge/string theory, the correspondence can be formulated mathematically more rigorous way. One of recent important achievements is instanton counting in four dimensional gauge theory by Nekrasov. In collaboration with T.Eguchi, we show that Nekrasov's partition function of instanton counting can be reproduced as a partition function of topological string theory. The partition function for SU(N) Seiberg-Witten theory is obtained by considering topological string whose target space is the ALE fibration of type A_{N-1} over P^1. Matter field can be incorporated by making a blow up on the target space. The methods used in this computation have close relations to various branches in mathematics, such as representation theory, combinatorics, knot theory and integrable systems.